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Continuity of Functions Defined by Lebesgue Integrals

Often times functions involving integrals are useful - especially in applied mathematics. We will now consider the various properties of functions defined by Lebesgue integrals. Let $I$ and $J$ be intervals and let $f : I \times J \to \mathbb{R}$ be a function. Then we can define a new, single variable function $F : J \to \mathbb{R}$ for all $y \in J$ by:

Proof of a): Suppose that conditions (1), (2), and (3) all hold. For each fixed $y \in J$, since $f_y$ is a measurable function on $I$ and $\mid f_y(x) \mid \leq g(x)$ almost everywhere on $I$, where $g$ is a Lebesgue integrable function, then we have from the Criterion for a Measurable Function to be Lebesgue Integrable page that $f_y$ is Lebesgue integrable for every $y \in J$. In other words, the following Lebesgue integral exists for all $y \in J$:

Proof of b) Now, fix $y \in J$ and let $(y_n)_{n=1}^{\infty}$ be any sequence of real numbers that converges to $y$, i.e., $\displaystyle{\lim_{n \to \infty} y_n = y}$. Now consider the sequence of functions $(f_{y_n})_{n=1}^{\infty}$. Each of these functions are Lebesgue integrable (as we proved above). Furthermore, $(f_{y_n})_{n=1}^{\infty}$ converges to $f_y$ almost everywhere on $I$ and additionally, the following inequality holds almost everywhere on $I$ and for all $n \in \mathbb{N}$:

Now, if $F$ is continuous at $y$ then from the Sequential Criterion for the Continuity of a Function we must have that $\lim_{n \to \infty} F(y_n) = F \left ( \lim_{n \to \infty} y_n \right ) = F(y)$. Evaluating the limit on the left and using $(*)$ and (3) gives us: